Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

TL;DR

This paper explores the nonlinear differential equations satisfied by certain classical modular forms, deriving generalized Chazy equations and related identities through a unified approach involving Picard-Fuchs equations and hypergeometric functions.

Contribution

It introduces a unified framework for deriving nonlinear differential equations for low-weight modular forms on various d0(N) groups, including generalized Chazy equations and transformation laws.

Findings

Derivation of coupled nonlinear differential equations for weight-1 modular forms.

Generation of divisor function and theta identities via q-expansions.

Establishment of transformation laws for elliptic integrals under d0(4).

Abstract

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under \Gamma_0(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard-Fuchs equations of triangle subgroups of PSL(2,R) which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with \Gamma(1) is treated.